Symbolically,—

PaM,SeM,⎱
MeS,⎱orMeS,⎰
orSeM,⎰ MiP,
⎯⎯⎯⎯
∴
or 		SeP
SoP		⎱
⎰		∴ PoS.

If it be required to retain the quantity of the original conclusion, that conclusion must be SoP, in this case then we have only two syllogisms fulfilling the given conditions.

355. Shew that if the proportion of B’s out of the class A is greater than that out of the class not-A, then the proportion 409 of A’s out of the class B will be greater than that out of the class not-B.[439] [J.]

[439] This and the following problem cannot properly be called problems on the syllogism. They are given as examples in numerical logic.

Let the number of A’s be denoted by N(A), the number of AB’s by N(AB), &c.
Then, since Every A is AB or Ab (by the law of excluded middle) and No A is both AB and Ab (by the law of contradiction), it follows that

N(A) = N(AB) + N(Ab).

We have to shew that

N(AB)N(Ab)
⎯⎯ > ⎯⎯
N(B)N(b)

follows from